Discrete Mathematics 2010-2018

Two most standard ways to measure the quality of a point set on the sphere are discrepancy and energy. In the former, one compares the proportion of points in certain subdomains to their area, while in the latter one views points as electrons that repel according to a certain force. We shall talk about various energy minimization problems on the sphere, when minimal energy induces uniform distribution, how the structure of the function affects minimizers, special point sets that arise as minimizers (tight frames, spherical designs), connections to spherical harmonics, Gegenbauer polynomials, and positive definite functions etc. Then we shall discus discrepancy on the sphere: known bounds, methods, constructions, as well as relations to energy.

This is the second lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subject involved in the talk will be assumed.

We investigate the model selection properties of the Lasso estimator in finite samples with no conditions on the regressor matrix X. We show that which covariates the Lasso estimator may potentially choose in high dimensions (where the number of explanatory variables p exceeds sample size n) depends only on X and the given penalization weights. This set of potential covariates can be determined through a geometric condition on X and may be small enough (less than or equal to n in cardinality) so that the Lasso estimator acts as a low-dimensional procedure also in high dimensions. Related to the geometric conditions in our considerations, we also provide a necessary and sufficient condition for uniqueness of the Lasso solutions.